4

Part of 1989 Poland - Second Round

Problems(1)

x_1a_1 + ...x_{11}a_{ 11} is divisible by 1989.

Source: Polish MO Recond Round 1989 p4

The given integers are

a_1, a_2, \ldots , a_{11}

. Prove that there exists a non-zero sequence

x_1, x_2, \ldots, x_{11}

with terms from the set

\{-1,0,1\}

such that the number

x_1a_1 + \ldots x_{11}a_{ 11}

is divisible by 1989.

number theorydividesdivisible